Calculus Examples

$\int_{1}^{3} (2 x^{3} - 4 x^{- 2}) d x$

Step 1

Remove parentheses.

$\int_{1}^{3} 2 x^{3} - 4 x^{- 2} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{1}^{3} 2 x^{3} d x + \int_{1}^{3} - 4 x^{- 2} d x$

Step 3

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{1}^{3} x^{3} d x + \int_{1}^{3} - 4 x^{- 2} d x$

Step 4

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$2 (\frac{1}{4} x^{4}]_{1}^{3}) + \int_{1}^{3} - 4 x^{- 2} d x$

Step 5

Combine $\frac{1}{4}$ and $x^{4}$ .

$2 (\frac{x^{4}}{4}]_{1}^{3}) + \int_{1}^{3} - 4 x^{- 2} d x$

Step 6

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$2 (\frac{x^{4}}{4}]_{1}^{3}) - 4 \int_{1}^{3} x^{- 2} d x$

Step 7

By the Power Rule, the integral of $x^{- 2}$ with respect to $x$ is $- x^{- 1}$ .

$2 (\frac{x^{4}}{4}]_{1}^{3}) - 4 (- x^{- 1}]_{1}^{3})$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $\frac{x^{4}}{4}$ at $3$ and at $1$ .

$2 ((\frac{3^{4}}{4}) - \frac{1^{4}}{4}) - 4 (- x^{- 1}]_{1}^{3})$

Step 8.2

Evaluate $- x^{- 1}$ at $3$ and at $1$ .

$2 (\frac{3^{4}}{4} - \frac{1^{4}}{4}) - 4 (- 3^{- 1} + 1^{- 1})$

Step 8.3

Simplify.

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Step 8.3.1

Raise $3$ to the power of $4$ .

$2 (\frac{81}{4} - \frac{1^{4}}{4}) - 4 (- 3^{- 1} + 1^{- 1})$

Step 8.3.2

One to any power is one.

$2 (\frac{81}{4} - \frac{1}{4}) - 4 (- 3^{- 1} + 1^{- 1})$

Step 8.3.3

Combine the numerators over the common denominator.

$2 \frac{81 - 1}{4} - 4 (- 3^{- 1} + 1^{- 1})$

Step 8.3.4

Subtract $1$ from $81$ .

$2 (\frac{80}{4}) - 4 (- 3^{- 1} + 1^{- 1})$

Step 8.3.5

Cancel the common factor of $80$ and $4$ .

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Step 8.3.5.1

Factor $4$ out of $80$ .

$2 \frac{4 \cdot 20}{4} - 4 (- 3^{- 1} + 1^{- 1})$

Step 8.3.5.2

Cancel the common factors.

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Step 8.3.5.2.1

Factor $4$ out of $4$ .

$2 \frac{4 \cdot 20}{4 (1)} - 4 (- 3^{- 1} + 1^{- 1})$

Step 8.3.5.2.2

Cancel the common factor.

$2 \frac{4 \cdot 20}{4 \cdot 1} - 4 (- 3^{- 1} + 1^{- 1})$

Step 8.3.5.2.3

Rewrite the expression.

$2 (\frac{20}{1}) - 4 (- 3^{- 1} + 1^{- 1})$

Step 8.3.5.2.4

Divide $20$ by $1$ .

$2 \cdot 20 - 4 (- 3^{- 1} + 1^{- 1})$

Step 8.3.6

Multiply $2$ by $20$ .

$40 - 4 (- 3^{- 1} + 1^{- 1})$

Step 8.3.7

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$40 - 4 (- \frac{1}{3} + 1^{- 1})$

Step 8.3.8

One to any power is one.

$40 - 4 (- \frac{1}{3} + 1)$

Step 8.3.9

Write $1$ as a fraction with a common denominator.

$40 - 4 (- \frac{1}{3} + \frac{3}{3})$

Step 8.3.10

Combine the numerators over the common denominator.

$40 - 4 \frac{- 1 + 3}{3}$

Step 8.3.11

Add $- 1$ and $3$ .

$40 - 4 (\frac{2}{3})$

Step 8.3.12

Combine $- 4$ and $\frac{2}{3}$ .

$40 + \frac{- 4 \cdot 2}{3}$

Step 8.3.13

Multiply $- 4$ by $2$ .

$40 + \frac{- 8}{3}$

Step 8.3.14

Move the negative in front of the fraction.

$40 - \frac{8}{3}$

Step 8.3.15

To write $40$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$40 \cdot \frac{3}{3} - \frac{8}{3}$

Step 8.3.16

Combine $40$ and $\frac{3}{3}$ .

$\frac{40 \cdot 3}{3} - \frac{8}{3}$

Step 8.3.17

Combine the numerators over the common denominator.

$\frac{40 \cdot 3 - 8}{3}$

Step 8.3.18

Simplify the numerator.

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Step 8.3.18.1

Multiply $40$ by $3$ .

$\frac{120 - 8}{3}$

Step 8.3.18.2

Subtract $8$ from $120$ .

$\frac{112}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{112}{3}$

Decimal Form:

$37.\overline{3}$

Mixed Number Form:

$37 \frac{1}{3}$

Step 10